A211795 - OEIS

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A211795

Number of (w,x,y,z) with all terms in {1,...,n} and w*x < 2*y*z.

203

0, 1, 11, 58, 177, 437, 894, 1659, 2813, 4502, 6836, 10008, 14121, 19449, 26117, 34372, 44422, 56597, 71044, 88160, 108115, 131328, 158074, 188773, 223604, 263172, 307719, 357715, 413493, 475690, 544480, 620632, 704381, 796413 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	COMMENTS	`Each sequence in the following guide counts 4-tuples` `(w,x,y,z) such that the indicated relation holds and the four numbers w,x,y,z are in {1,...,n}. The notation "m div" means that m divides every term of the sequence.` `A211058 ... wx <= yz` `A211787 ... wx <= 2yz` `A211795 ... wx < 2yz` `A211797 ... wx > 2yz` `A211809 ... wx >= 2yz` `A211812 ... wx <= 3yz` `A211917 ... wx < 3yz` `A211918 ... wx > 3yz` `A211919 ... wx >= 3yz` `A211920 ... 2wx < 3yz` `A211921 ... 2wx <= 3yz` `A211922 ... 2wx > 3yz` `A211923 ... 2wx >= 3yz` `A212019 ... wx = 2yz ..... 2 div` `A212020 ... wx = 3yz ..... 2 div` `A212021 ... 2wx = 3yz .... 2 div` `A212047 ... wx = 4yz` `A212048 ... 3wx = 4yz .... 2 div` `A212049 ... wx = 5yz ..... 2 div` `A212050 ... 2wx = 5yz .... 2 div` `A212051 ... 3wx = 5yz .... 2 div` `A212052 ... 4wx = 5yz .... 2 div` `A209978 ... wx = yz + 1 .. 2 div` `A212053 ... wx <= yz + 1` `A212054 ... wx > yz + 1` `A212055 ... wx <= yz + 2` `A212056 ... wx > yz + 2` `A197168 ... wx = yz + 2 .. 2 div` `A061201 ... w = xyz` `A212057 ... w < xyz` `A212058 ... w >= xyz` `A212059 ... w = xyz - 1` `A212060 ... w = xyz - 2` `A212061 ... wx = (yz)^2` `A212062 ... w^2 = xyz` `A212063 ... w^2 < xyz` `A212064 ... w^2 >= xyz` `A212065 ... w^2 <= xyz` `A212066 ... w^2 > xyz` `A212067 ... w^3 = xyz` `A002623 ... w = 2x + y + z` `A006918 ... w = 2x + 2y + z` `A000601 ... w = x + 2y + 3z (except for initial 0's)` `A212068 ... 2w = x + y + z` `A212069 ... 3w = x + y + z (w = average{x,y,z})` `A212088 ... 3w < x + y + z` `A212089 ... 3w >= x + y + z` `A212090 ... w < x + y + z` `A000332 ... w >= x + y + z` `A212145 ... w < 2x + y + z` `A001752 ... w >= 2x + y + z` `A001400 ... w = 2x +3y + 4z` `A005900 ... w = -x + y + z` `A192023 ... w = -x + y + z + 2` `A212091 ... w^2 = x^2 + y^2 + z^2 ... 3 div` `A212087 ... w^2 + x^2 = y^2 + z^2` `A212092 ... w^2 < x^2 + y^2 + z^2` `A212093 ... w^2 <= x^2 + y^2 + z^2` `A212094 ... w^2 > x^2 + y^2 + z^2` `A212095 ... w^2 >= x^2 + y^2 + z^2` `A212096 ... w^3 = x^3 + y^3 + z^3 ... 6 div` `A212097 ... w^3 < x^3 + y^3 + z^3` `A212098 ... w^3 <= x^3 + y^3 + z^3` `A212099 ... w^3 > x^3 + y^3 + z^3` `A212100 ... w^3 >= x^3 + y^3 + z^3` `A212101 ... wx^2 = yz^2` `A212102 ... 1/w = 1/x + 1/y + 1/z` `A212103 ... 3/w = 1/x + 1/y + 1/z; w = h.m. of {x,y,z}` `A212104 ... 3/w >= 1/x + 1/y + 1/z; w >= h.m.` `A212105 ... 3/w < 1/x + 1/y + 1/z; w < h.m.` `A212106 ... 3/w > 1/x + 1/y + 1/z; w > h.m.` `A212107 ... 3/w <= 1/x + 1/y + 1/z; w <= h.m.` `A212133 ... median(w,x,y,z) = mean(w,x,y,z)` `A212134 ... median(w,x,y,z) <= mean(w,x,y,z)` `A212135 ... median(w,x,y,z) > mean(w,x,y,z)` `A212241 ... wx + yz > n` `A212243 ... 2wx + yz = n` `A212244 ... w = xyz - n` `A212245 ... w = xyz - 2n` `A212246 ... 2w = x + y + z - n` `A212247 ... 3w = x + y + z + n` `A212249 ... 3w < x + y + z + n` `A212250 ... 3w >= x + y + z + n` `A212251 ... 3w = x + y + z + n + 1` `A212252 ... 3w = x + y + z + n + 2` `A212254 ... w = x + 2y + 3z - n` `A212255 ... w^2 = mean(x^2, y^2, z^2)` `A212256 ... 4/w = 1/x + 1/y +1/z + 1/n` `In the list above, if the relation in the second column is of the form "w rel ax + by + cz" then the sequence is linearly recurrent. In the list below, the same is true for expressions involving more than one relation.` `A000332 ... w < x <= y < z .... C(n,4)` `A000914 ... w < x <= y < z .... Stirling 1st kind` `A000914 ... w < x <= y >= z ... Stirling 1st kind` `A050534 ... w < x < y >= z .... tritriangular` `A001296 ... w <= x <= y >= z .. 4-dim pyramidal` `A006322 ... x < x > y >= z` `A002418 ... w < x >= y < z` `A050534 ... w < x >=y >= z` `A212415 ... w < x >= y <= z` `A001296 ... w < x >= y <= z` `A212246 ... w <= x > y <= z` `A006322 ... w <= x >= y <= z` `A212501 ... w > x < y >= z` `A212503 ... w < 2x and y < 2z ..... A (note below)` `A212504 ... w < 2x and y > 2z ..... A` `A212505 ... w < 2x and y >= 2z .... A` `A212506 ... w <= 2x and y <= 2z ... A` `A212507 ... w < 2x and y <= 2z .... B` `A212508 ... w < 2x and y < 3z ..... C` `A212509 ... w < 2x and y <= 3z .... C` `A212510 ... w < 2x and y > 3z ..... C` `A212511 ... w < 2x and y >= 3z .... C` `A212512 ... w <= 2x and y < 3z .... C` `A212513 ... w <= 2x and y <= 3z ... C` `A212514 ... w <= 2x and y > 3z .... C` `A212515 ... w <= 2x and y >= 3z ... C` `A212516 ... w > 2x and y < 3z ..... C` `A212517 ... w > 2x and y <= 3z .... C` `A212518 ... w > 2x and y > 3z ..... C` `A212519 ... w > 2x and y >= 3z .... C` `A212520 ... w >= 2x and y < 3z .... C` `A212521 ... w >= 2x and y <= 3z ... C` `A212522 ... w >= 2x and y > 3z .... C` `A212523 ... w + x < y + z` `A212560 ... w + x <= y + z` `A212561 ... w + x = 2y + 2z` `A212562 ... w + x < 2y + 2z ....... B` `A212563 ... w + x <= 2y + 2z ...... B` `A212564 ... w + x > 2y + 2z ....... B` `A212565 ... w + x >= 2y + 2z ...... B` `A212566 ... w + x = 3y + 3z` `A212567 ... 2w + 2x = 3y + 3z` `A212570 ... \|w - x\| = \|x - y\| + \|y - z\|` `A212571 ... \|w - x\| < \|x - y\| + \|y - z\| ... B ... 4 div` `A212572 ... \|w - x\| <= \|x - y\| + \|y - z\| .. B` `A212573 ... \|w - x\| > \|x - y\| + \|y - z\| ... B ... 2 div` `A212574 ... \|w - x\| >= \|x - y\| + \|y - z\| .. B` `A212575 ... 2\|w - x\| = \|x - y\| + \|y - z\|` `A212576 ... \|w - x\| = 2\|x - y\| + 2\|y - z\|` `A212577 ... \|w - x\| = 2\|x - y\| - \|y - z\|` `A212578 ... 2\|w - x\| = \|x - y\| - \|y - z\|` `A212579 ... min{\|w-x\|,\|w-y\|} = min{\|x-y\|,\|x-z\|}` `A212692 ... w = \|x - y\| + \|y - z\| ............... 2 div` `A212568 ... w < \|x - y\| + \|y - z\| ............... 2 div` `A212573 ... w <= \|x - y\| + \|y - z\| .............. 2 div` `A212574 ... w > \|x - y\| + \|y - z\|` `A212575 ... w >= \|x - y\| + \|y - z\|` `A212676 ... w + x = \|x - y\| + \|y - z\| ......... H` `A212677 ... w + y = \|x - y\| + \|y - z\|` `A212678 ... w + x + y = \|x - y\| + \|y - z\|` `A006918 ... w + x + y + z = \|x - y\| + \|y - z\| . H` `A212679 ... \|x - y\| = \|y - z\| ................. H` `A212680 ... \|x - y\| = \|y - z\| + 1 ..............H 2 div` `A212681 ... \|x - y\| < \|y - z\| ................... 2 div` `A212682 ... \|x - y\| >= \|y - z\|` `A212683 ... \|x - y\| = w + \|y - z\| ............... 2 div` `A212684 ... \|x - y\| = n - w + \|y - z\|` `A212685 ... \|w - x\| = w + \|y - z\|` `A186707 ... \|w - x\| < w + \|y - z\| ... (Note D)` `A212714 ... \|w - x\| >= w + \|y - z\| .......... H . 2 div` `A212686 ... 2\|w - x\| = n + \|y - z\| ............. 4 div` `A212687 ... 2\|w - x\| < n + \|y - z\| ......... B` `A212688 ... 2\|w - x\| < n + \|y - z\| ......... B . 2 div` `A212689 ... 2\|w - x\| > n + \|y - z\| ......... B . 2 div` `A212690 ... 2\|w - x\| <= n + \|y - z\| ........ B` `A212691 ... w + \|x - y\| = \|x - z\| + \|y - z\| . E . 2 div` `...` `In the above lists, all the terms of (w,x,y,z) are in {1,...,n}, but in the next lists they are all in {0,...,n}, and sequences are all linearly recurrent.` `R=range{w,x,y,z}=max{w,x,y,z}-min{w,x,y,z}.` `A212740 ... max{w,x,y,z} < 2min{w,x,y,z} .... A` `A212741 ... max{w,x,y,z} >= 2min{w,x,y,z} ... A` `A212742 ... max{w,x,y,z} <= 2min{w,x,y,z} ... A` `A212743 ... max{w,x,y,z} > 2min{w,x,y,z} .... A . 2 div` `A212744 ... w=range (=max-min) ............... E` `A212745 ... w=max{w,x,y,z} - 2min{w,x,y,z}` `A212746 ... R is in {w,x,y,z} ................ E` `A212569 ... R is not in {w,x,y,z} ............ E` `A212749 ... w=R or x<R or y<R or z<R ......... A` `A212750 ... w=R or x=R or y<R or z<R ......... A` `A212751 ... w=R or x=R or y<R or z<R ......... A` `A212752 ... w<R or x<R or y<R or z>R ......... A` `A212753 ... w<R or x<R or y>R or z>R ......... D` `A212754 ... w<R or x>R or y>R or z>R ......... D` `A002415 ... w = x + R ........................ D` `A212755 ... \|w - x\| = R ...................... D` `A212756 ... 2w = x + R` `A212757 ... 2w = R` `A212758 ... w = floor(R/2)` `A002413 ... w = floor((x+y+z/2))` `A212759 ... w, x, y are even` `A212760 ... w is even and x = y + z .......... E` `A212761 ... w is odd and x and y are even .... F . 2 div` `A212762 ... w and x are odd y is even ........ F . 2 div` `A212763 ... w, x, y are odd .................. F` `A212764 ... w, x, y are even and z is odd .... F` `A030179 ... w and x are even and y and z odd` `A212765 ... w is even and x,y,z are odd ...... F` `A212766 ... w is even and x is odd ........... A . 2 div` `A212767 ... w and x are even and w+x=y+z ..... E` `A212889 ... R is even ........................ A` `A212890 ... R is odd ......................... A . 2 div` `A212742 ... w-x, x-y, y-z are all even ....... A` `A212892 ... w-x, x-y, y-z are all odd ........ A` `A212893 ... w-x, x-y, y-z have same parity ... A` `A005915 ... min{\|w-x\|, \|x-y\|, \|y-z\|} = 0` `A212894 ... min{\|w-x\|, \|x-y\|, \|y-z\|} = 1` `A212895 ... min{\|w-x\|, \|x-y\|, \|y-z\|} = 2` `A179824 ... min{\|w-x\|, \|x-y\|, \|y-z\|} > 0` `A212896 ... min{\|w-x\|, \|x-y\|, \|y-z\|} <= 1` `A212897 ... min{\|w-x\|, \|x-y\|, \|y-z\|} > 1` `A212898 ... min{\|w-x\|, \|x-y\|, \|y-z\|} <= 2` `A212899 ... min{\|w-x\|, \|x-y\|, \|y-z\|} > 2` `A212901 ... \|w-x\| = \|x-y\| = \|y-z\|` `A212900 ... \|w-x\|, \|x-y\|, \|y-z\| are distinct . G` `A212902 ... \|w-x\| < \|x-y\| < \|y-z\| ............ G` `A212903 ... \|w-x\| <= \|x-y\| <= \|y-z\| .......... G` `A212904 ... \|w-x\| + \|x-y\| + \|y-z\| = n ........ H` `A212905 ... \|w-x\| + \|x-y\| + \|y-z\| = 2n ....... H` `...` `Note A: A212503-A212506 (and others) have these recurrence coefficients: 2,2,-6,0,6,-2,-2,1.` `B: 3,-1,-5,5,1,-3,1` `C: 0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1` `D: 4,-5,0,5,-4,1` `E: 1,3,-3,-3,3,1,-1` `F: 1,4,-4,-6,6,4,-4,-1,1` `G: 2,1,-3,-1,1,3,-1,-2,1` `H: 2,1,-4,1,2,-1`
	REFERENCES	`A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.` `P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.`
	LINKS	`Bo Gyu Jeong, Table of n, a(n) for n = 0..200`
	FORMULA	`a(n) = n^4 - A211809(n).`
	EXAMPLE	`a(2)=11 counts these (w,x,y,z): (1,1,1,1), (1,1,1,2), (1,1,2,1), (2,1,2,1), (2,1,1,2), (1,2,2,1), (1,2,1,2), (1,1,2,2), (1,2,2,2), (2,1,2,2), (2,2,2,2).`
	MATHEMATICA	`t = Compile[{{n, _Integer}}, Module[{s = 0},` `(Do[If[wx < 2 yz, s = s + 1], {w, 1, #},` `{x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];` `Map[t[#] &, Range[0, 40]] (* A211795 )` `( Peter J. C. Moses, Apr 13 2012 *)`
	CROSSREFS	`Cf. A000583 (n^4), A210000, A211809, A212959.` `Sequence in context: A048366 A107425 A211921 * A256226 A290360 A359719` `Adjacent sequences: A211792 A211793 A211794 * A211796 A211797 A211798`
	KEYWORD	`nonn`
	AUTHOR	`Clark Kimberling, Apr 27 2012`
	STATUS	`approved`

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Last modified April 27 16:27 EDT 2024. Contains 372020 sequences. (Running on oeis4.)